The figure below shows the top-down view of a disk on a vertical axle. The disk can rotate around the axle with a natural frequency of ω rad.s-1.

There is a spring attached to the periphery of the disk with a spherical mass at the end of it. The entire collection rests on a frictionless surface so that the sphere can only move horizontally.

The horizontal distance from the centre of the disk to the centre of the sphere is shown as "x" in the diagram below. When the disk is at rest:

x = x0 = rdisk + xS0 + rsphere

Where rdisk and r sphere are the respective radii, and xS0is the unstretched length of the spring.

As the disk spins at faster speeds, the spring will extend and the sphere will move further away from the centre of the disk:

x = x0 + Δx

Draw a free-body diagram of the mass, showing all forces that are acting on it, for the situation where the disc-mass system is rotating at a constant angular velocity.

Hence write the equations of motion for the mass for all 3 axes.

The disc-spring-mass system has the following characteristics:

mass = 1 kg (for the spherical mass at the end of the spring),

k= 140 N.m-1,

x0 = 1.2 m when the system is stationary.

Calculate the angular velocity, ω, that will result in an extension of 0.3 m (i.e. x = 1.5 m).

Give your answer in rad.s-1 to at least 3 significant figures

Answer:

The sphere has a radius of 0.1 m. Calculate the moment of inertia of the sphere around the axis at the centre of the disc for the above situation.

Give your answer in kg.m2 to at least 3 significant figures

Answer: